A055010 a(0) = 0; for n > 0, a(n) = 3*2^(n-1) - 1.

0, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 0

Author

Henry Bottomley, May 31 2000

Keywords

• easy
• nonn

Comments

Apart from leading term (which should really be 3/2), same as A083329.

Written in binary, a(n) is 1011111...1.

The sequence 2, 5, 11, 23, 47, 95, ... apparently gives values of n such that Nim-factorial(n) = 2. Cf. A059970. However, compare A060152. More work is needed! - John W. Layman, Mar 09 2001

With offset 1, number of (132,3412)-avoiding two-stack sortable permutations.

Number of descents after n+1 iterations of morphism A007413.

a(n) = A164874(n,1), n>0; subsequence of A030130. - Reinhard Zumkeller, Aug 29 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-1). - Milan Janjic, Jan 24 2010

a(n) is the total number of records over all length n binary words. A record in a word a_1,a_2,...,a_n is a letter a_j that is larger than all the preceding letters. That is, a_j>a_i for all iGeoffrey Critzer, Jul 18 2020

Called Thabit numbers after the Syrian mathematician Thābit ibn Qurra (826 or 836 - 901). - Amiram Eldar, Jun 08 2021

a(n) is the number of objects in a pile that represents a losing position in a Nim game, where a player must select at least one object but not more than half of the remaining objects, on their turn. - Kiran Ananthpur Bacche, Feb 03 2025

Examples

			a(3) = 3*2^2 - 1 = 3*4 - 1 = 11.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric S. Egge and Toufik Mansour, 132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers, arXiv:math/0205206 [math.CO], 2002.
• Sergey Kitaev and Toufik Mansour, Counting the occurrences of generalized patterns in words generated by a morphism, arXiv:math/0210170 [math.CO], 2002.
• Hunar Sherzad Taher and Saroj Kumar Dash, On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 448-459. See p. 2.
• Eric Weisstein's World of Mathematics, Thabit ibn Kurrah Number.
• Wikipedia, Thabit number.
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. A007505 for primes in this sequence. Apart from initial term, same as A052940 and A083329.

Cf. A266550 (independence number of the n-Mycielski graph).

Cf. A000079, A000225, A007283, A007413, A010036, A030130, A033484, A059970, A060152, A099258, A118654, A164874, A196168.

Programs

GAP

Concatenation([0], List([1..35], n-> 3*2^(n-1)-1)); # G. C. Greubel, May 06 2019

Magma

[Floor(3*2^(n-1) - 1): n in [0..35]]; // Vincenzo Librandi, May 18 2011

Mathematica

Join[{0},3*2^Range[0,34]-1] (* Harvey P. Dale, May 05 2013 *)

PARI

a(n)=3*2^n\2 - 1 \\ Charles R Greathouse IV, Apr 08 2016

Sage

[0]+[3*2^(n-1)-1 for n in (1..35)] # G. C. Greubel, May 06 2019

Formula

a(n) = A118654(n-1, 4), for n > 0.

a(n) = 2*a(n-1) + 1 = a(n-1) + A007283(n-1) = A007283(n)-1 = A000079(n) + A000225(n + 1) = A000079(n + 1) + A000225(n) = 3*A000079(n) - 1 = 3*A000225(n) + 2.

a(n) = A010036(n)/2^(n-1). - Philippe Deléham, Feb 20 2004

a(n) = A099258(A033484(n)-1) = floor(A033484(n)/2). - Reinhard Zumkeller, Oct 09 2004

G.f.: x*(2-x)/((1-x)*(1-2*x)). - Philippe Deléham, Oct 04 2011

a(n+1) = A196168(A000079(n)). - Reinhard Zumkeller, Oct 28 2011

E.g.f.: (3*exp(2*x) - 2*exp(x) - 1)/2. - Stefano Spezia, Sep 14 2024